Univrsiy of Wisconsin Milwauk UWM Digial Commons Thss and Dissraions Augus 25 Transin Prformanc Analysis of Srial Producion Lins Yang Sun Univrsiy of Wisconsin-Milwauk Follow his and addiional works a: hp://dc.uwm.du/d Par of h Indusrial Enginring Commons Rcommndd Ciaion Sun, Yang, "Transin Prformanc Analysis of Srial Producion Lins" (25). Thss and Dissraions. Papr 23. This Thsis is brough o you for fr and opn accss by UWM Digial Commons. I has bn accpd for inclusion in Thss and Dissraions by an auhorizd adminisraor of UWM Digial Commons. For mor informaion, plas conac krisinw@uwm.du.
TRANSIENT PERFORMANCE ANALYSIS OF SERIAL PRODUCTION LINES by Yang Sun A Thsis Submid in Parial Fulfillmn of h Rquirmns for h Dgr of Masr of Scinc in Enginring a Th Univrsiy of Wisconsin-Milwauk Augus 25
ABSTRACT TRANSIENT PERFORMANCE ANALYSIS OF SERIAL PRODUCTION LINES by Yang Sun Th Univrsiy of Wisconsin-Milwauk, 25 Undr h Suprvision of Profssor Liang Zhang Producion lins wih unrliabl machins and fini buffrs ar characrizd by boh sady-sa prformanc and ransin bhavior. Th sady-sa prformanc has bn analyzd xnsivly for ovr 5 yars. Transin bhavior, howvr, is rarly sudid and rmains lss xplord. In pracic, a lo of h ral producion sysms ar running parially or nirly in ransin priods. Thrfor, ransin analysis is of significan pracical imporanc. Mos of h pas rsarch on producion sysms focuss on discr marials flow which uiliis Markov chain analysis. This dissraion is dvod o invsiga h ffcs of sysm paramrs on prformanc masurs for ransin srial producion lin wih ohr machin rliabiliy modls. Th rliabiliy modls invsigad in his dissraion ar xponnial and no-xponnial (,, ). In a ral producion lin sysm, machin rliabiliy modls ar much mor difficul o idnify. Sricly spaking, i rquirs h idnificaions of h hisograms of up- and downim, which rquirs a vry larg numbr of masurmns during a long priod of ii
im. Th rsul may b ha h machins ral rliabiliy modl on h facory floor ar, pracically, nvr known. Thrfor, i is of significan pracical imporanc o invsiga h gnral ffcs of sysm paramrs on prformanc masurs for ransin srial producion lin wih diffrn rliabiliy modls. Th sysm paramrs includ machin fficincy, raio of N and T down (), machins avrag downim T down, and cofficin of variaion CV. Th prformanc masurs includ sling im of producion ra ( ), sling im of work-in-procss ( ), oal producion (TP), producion loss (PL). Th rlaionship bwn h prformanc masurs and sysm paramrs rvals h fundamnal principls ha characriz h bhavior of such sysms and can b usd as a guidlin for produc lins managmn and improvmn. Mos prvious rsarch sudis ar limid o wo or hr machin sysm du o h chnical complxiy. Furhrmor, prsnly hr ar no analyical ools o addrss h problms wih mulipl machins and buffrs during ransin priods. This dissraion addrsss his problm by using simulaions wih C++ programming o valua h mulipl machins (up o ) and buffrs and dmonsra h ransin prformanc a diffrn condiions. Th simulaion mhod dos no only provid quanifid ransin prformanc rsuls for a givn producion lin, bu also provids a valuabl ool o invsiga h sysm paramr ffcs and how o manag and improv h xising producion lin. iii
Copyrigh by Yang Sun, 25 All Righs Rsrvd
To my parns, my husband and my son
TABLE OF CONTENTS LIST OF FIGURES... viii LIST OF TABLES... xi Chapr Inroducion.... Moivaion....2 Oulin...4 Chapr 2 Sysm Modls and Problms...5 2. Trminology...5 2.2 Machin Rliabiliy Modls...7 2.3 Sysms Considrd... 2.3. Coninuous Srial Producion Lins... 2.3.2 Sysms Considrd...2 2.3.3 Sysm Paramrs...3 2.4 Prformanc Masurs...4 2.5 Problm Samn...5 Chapr 3 Exponnial Sysms...7 3. Transins of Exponnial Lins...7 3.. Transins of Individual Exponnial Machin...7 3..2 Transins of Buffrlss Exponnial Srial Lin...9 3..3 Exponnial vs. Gomric...2 3.2 Transin Prformanc Analysis...28 3.2. Effcs of...28 3.2.2 Effcs of...3 3.2.3 Effcs of T down...3 3.3 Sling Tim...32 3.3. Effcs of...32 3.3.2 Effcs of...34 3.3.3 Effcs of T down...36 3.4 Toal Producion...38 3.4. Effcs of...38 3.4.2 Effcs of...4 3.4.3 Effcs of Tdown...4 3.5 Producion Loss...42 3.5. Effcs of...43 3.5.2 Effcs of...44 3.5.3 Effcs of T down...46 3.6 Summary...47 Chapr 4,, Log-Normal Sysms...5
4. Transin Prformanc Analysis...5 4.. Effcs of...5 4..2 Effcs of...54 4..3 Effcs of T down...55 4..4 Effcs of CV up...57 4..5 Effcs of CV down...58 4.2 Sling Tim...6 4.2. Effcs of...6 4.2.2 Effcs of...62 4.2.3 Effcs of T down...64 4.2.4 Effcs of CV up...66 4.2.5 Effcs of CV down...67 4.3 Toal Producion...69 4.3. Effcs of...69 4.3.2 Effcs of...7 4.3.3 Effcs of T down...73 4.3.4 Effcs of CV up...75 4.3.5 Effcs of CV down...77 4.4 Producion Loss...79 4.4. Effcs of...79 4.4.2 Effcs of...8 4.4.3 Effcs of T down...83 4.4.4 Effcs of CV up...85 4.4.5 Effcs of CV down...86 4.5 Summary...88 Chapr 5 Conclusions and Fuur Works...9 Bibliography...95 vii
LIST OF FIGURES Figur Srial producion lin...5 Figur 2 Exponnial Rliabiliy Modl...8 Figur 3 Srucural Modl... Figur 4 Transins of machin sa of an individual xponnial machin (T up = 8, T down = 2)...9 Figur 5 Transins of buffrlss fiv-machin xponnial lin wih idnical machins (T up = 8, T down = 2, machins iniially down)...2 Figur 6 Transins of buffrlss fiv-machin xponnial lin wih idnical machins (T up = 8, T down = 2, machins iniially up)...2 Figur 7. Approximaion of PR() of fiv-machin xponnial lin wih idnical machins and idnical buffrs (T up = 4, T down =, N =, all machins iniially down, all buffrs iniially mpy)...23 Figur 8 Approximaion of CR() of fiv-machin xponnial lin wih idnical machins and idnical buffrs (T up = 4, T down =, N =, all machins iniially down, all buffrs iniially mpy)...24 Figur 9 Approximaion of WIPi() of fiv-machin xponnial lin wih idnical machins and idnical buffrs (T up = 4, T down =, N =, all machins iniially down, all buffrs iniially mpy)...25 Figur Approximaion of STi() of fiv-machin xponnial lin wih idnical machins and idnical buffrs (T up = 4, T down =, N =, all machins iniially down, all buffrs iniially mpy)...26 Figur Approximaion of BLi() of fiv-machin xponnial lin wih idnical machins and idnical buffrs (T up = 4, T down =, N =, all machins iniially down, all buffrs iniially mpy)...27 Figur 2 Effcs of on Transin Prformanc of Exponnial Lins...29 Figur 3 Effcs of on Transin Prformanc of Exponnial Lins...3 Figur 4 Effcs of T down on Transin Prformanc of Exponnial Lins...32 Figur 5 Effcs of on spr and swip of Exponnial Lins...34 viii
Figur 6 Effcs of on spr and swip of Exponnial Lins...36 Figur 7 Effcs of T down on spr and swip of Exponnial Lins...37 Figur 8 Effcs of on Toal Producion of Exponnial Lins...39 Figur 9 Effcs of on Toal Producion of Exponnial Lins...4 Figur 2 Effcs of T down on Toal Producion of Exponnial Lins...42 Figur 2 Effcs of on Producion Loss of Exponnial Lins...44 Figur 22 Effcs of on Producion Loss of Exponnial Lins...45 Figur 23 Effcs of T down on Producion Loss of Exponnial Lins...47 Figur 24 Effcs of in Transin Procss of Non- Exponnial Lins...53 Figur 25 Effcs of in Transin Procss of Non- Exponnial Lins...55 Figur 26 Effcs of T down in Transin Procss of Non- Exponnial Lins...57 Figur 27 Effcs of CV up in Transin Procss of Non- Exponnial Lins...58 Figur 28 Effcs of CV down in Transin Procss of Non- Exponnial Lins...6 Figur 29 Effcs of on spr and swip of Non-Exponnial Lins...62 Figur 3 Effcs of on spr and swip of Non-Exponnial Lins...64 Figur 3 Effcs of T down on spr and swip of Non-Exponnial Lins...65 Figur 32 Effcs of CV up on s PR and swip of Non-Exponnial Lins...67 Figur 33 Effcs of CV down on spr and swip of Non-Exponnial Lins...68 Figur 34 Effcs of on Toal Producion of Non-Exponnial Lins...7 Figur 35 Effcs of on Toal Producion of Non-Exponnial Lins...72 Figur 36 Effcs of T down on Toal Producion of Non-Exponnial Lins...74 Figur 37 Effcs of CV up on Toal Producion of Non-Exponnial Lins...76 Figur 38 Effcs of CV down on Toal Producion of Non-Exponnial Lins...78 ix
Figur 39 Effcs of on Producion Loss of Non-Exponnial Lins...8 Figur 4 Effcs of on Producion Loss of Non-Exponnial Lins...82 Figur 4 Effcs of T down on Producion Loss of Non-Exponnial Lins...84 Figur 42 Effcs of CV up on Producion Loss of Non-Exponnial Lins...86 Figur 43 Effcs of CV down on Producion Loss of Non-Exponnial Lins...88 x
LIST OF TABLES Tabl Expcaion, varianc, and cofficins of variaion of coninuous random variabls... Tabl 2 Exponnial Sysm Transin Prformanc Analysis Valus...28 Tabl 3 Exponnial Sysm Paramrs (Effcs of on ransin prformanc)...29 Tabl 4 Exponnial Sysm Paramrs (Effcs of on ransin prformanc)...3 Tabl 5 Exponnial Sysm Paramrs (Effcs of T down on ransin prformanc)...3 Tabl 6 Exponnial Sysm Paramrs (Effcs of on spr and swip )...32 Tabl 7 Exponnial Sysm Paramrs (Effcs of on spr and swip )...34 Tabl 8 Exponnial Sysm Paramrs (Effcs of T down on spr and swip )...37 Tabl 9 Exponnial Sysm Paramrs (Effcs of on TP)...38 Tabl Exponnial Sysm Paramrs (Effcs of on TP)...4 Tabl Exponnial Sysm Paramrs (Effcs of T down on TP)...4 Tabl 2 Exponnial Sysm Paramrs (Effcs of on PL)...43 Tabl 3 Exponnial Sysm Paramrs (Effcs of on PL)...44 Tabl 4 Exponnial Sysm Paramrs (Effcs of T down on PL)...46 Tabl 5 Effcs of sysm paramrs (,, and T down ) on h spr and swip, TP and PL48 Tabl 6 Non-Exponnial Sysm Transin Prformanc Analysis Valus...5 Tabl 7 Non-Exponnial Sysm Paramrs( Effcs of on ransin prformanc)..52 Tabl 8 Non-Exponnial Sysm Paramrs (Effcs of on ransin prformanc).54 Tabl 9 Non-Exponnial Sysm Paramrs (Effcs of T down on ransin prformanc)55 Tabl 2 Non-Exponnial Sysm Paramrs (Effcs of CV up on ransin prformanc)57 xi
Tabl 2 Non-Exponnial Sysm Paramrs (Effcs of CV down on ransin prformanc)...58 Tabl 22 Non-Exponnial Sysm Paramrs (Effcs of on spr and swip )...6 Tabl 23 Non-Exponnial Sysm Paramrs (Effcs of on spr and swip )...62 Tabl 24 Non-Exponnial Sysm Paramrs (Effcs of T down on spr and swip )...64 Tabl 25 Non-Exponnial Sysm Paramrs (Effcs of CV up on spr and swip )...66 Tabl 26 Non-Exponnial Sysm Paramrs (Effcs of CV down on spr and swip )...67 Tabl 27 Non-Exponnial Sysm Paramrs (Effcs of on TP)...69 Tabl 28 Non-Exponnial Sysm Paramrs (Effcs of on TP)...7 Tabl 29 Non-Exponnial Sysm Paramrs (Effcs of T down on TP)...73 Tabl 3 Non-Exponnial Sysm Paramrs (Effcs of CV up on TP)...75 Tabl 3 Non-Exponnial Sysm Paramrs (Effcs of CV down on TP)...77 Tabl 32 Non-Exponnial Sysm Paramrs (Effcs of on PL)...79 Tabl 33 Non-Exponnial Sysm Paramrs (Effcs of on PL)...8 Tabl 34 Non-Exponnial Sysm Paramrs (Effcs of T down on PL)...83 Tabl 35 Non-Exponnial Sysm Paramrs (Effcs of CV up on PL)...85 Tabl 36 Non-Exponnial Sysm Paramrs (Effcs of CV down on PL)...86 Tabl 37 Effcs of sysm paramrs (,, and T down ) on h spr and swip, TP and PL...89 xii
ACNOWLEDGMENTS Firs and formos, I would lik o xprss my dps graiud o my advisor, Dr. Liang Zhang, for his guidanc and coninuous suppor hroughou my masr sudy. Dr. Zhang no only passd m knowldg bu also augh m how o hink and solv problms in my rsarch. Wihou his suppor and guidanc, i would no hav bn possibl for m o compl his dissraion. I is him who brough m ino h fascinaing world of producion sysm nginring. I hav bnfid rmndously from his vision, chnical insigh and profssionalism. His ddicad rsarch aliud dply influncs my sudy and rsarch. I am graful o all my commi mmbrs for hir hlp and suppor. I apprcia Dr. Wilkisar Oino and Dr. Xiang Fang for hir valuabl im and srving my commi. I am hankful o my collagu, Guorong Chn, for hlping m solving los of difficulis during my sudy. Finally, I would lik o ddica his hsis o my parns, Bojun Sun and Yuhua Yang, my husband, Tifu Zhao, and my son, Ehan Zhao. Thir suppor and lov mak my masr sudy a happy journy. xiii
Chapr Inroducion. Moivaion Producion lins wih unrliabl machins and fini buffrs ar characrizd by boh sady-sa prformanc and ransin bhavior. Th sady-sa prformanc has bn analyzd xnsivly for ovr 5 yars [-8]. In conras, ransin bhavior is lss sudid in h pas. Acually a lo of sysms ar running parially or nirly in ransin priod. For insanc, in som sysms, buffrs will b purgd a h nd of a shif and hrfor h sysms will bgin producion undr mpy buffr condiion. In ohr sysms, machins can sar or shu down a diffrn im in which cas sysms ar also running in ransin priod. Thrfor, in a manufacuring nvironmn, producion ransins, i.., h procssing im o rach sady sa, ar of significan pracical imporanc. If h sady sa is rachd afr a rlaivly long priod of im, h sysm may suffr subsanial producion losss. For insanc, i has bn shown ha if h cycl im of a producion sysm is minu and h plan shif is 5 minus, h sysm may los mor han % of is producion du o ransins, if a h bginning of h shif all buffrs wr mpy [9]. Thrfor, ransin analysis in producion lins is indispnsabl for a pracical producion sysm. Dspi h imporanc of ransin analysis, ransin prformanc is lss sudid and sill rmain unxplord in liraurs. Among h rviwd liraurs, prformanc analysis of srial producion lins wih Brnoulli machins during ransins hav bn
2 discussd in [9]. Thy invsigad propris of ransins of producion ra and work-in-procss for Brnoulli machin lins by using analyical mhod which is Markov chain analysis. I is shown ha h ransins of producion ra and work-in-procss ar drmind by h scond largs ignvalu of h ransiion marix of h associa Markov chain and h pr-xponnial facor. Th sling im and producion loss du o ransins ar also analyzd. To avoid h producion loss during ransins, i is suggsd ha all buffrs ar iniially a las half full. On h ohr hand, mos of h pas rsarch on producion sysms focuss on discr marials flow which uiliis Markov chain analysis [,, 2]. Thr is an incrasing numbr of rsarch on producion lins wih coninuous marials flow. Among hm, a complmnary sudy of coninuous marials flow producion sysms has bn conducd [3]. Th hroughpu and bolnck in assmbly sysms wih nonxponnial machins ar also sudid [4]. Howvr, i is mosly assumd ha h im o failur and h im o rpair ar xponnially disribud or drminisic. For insanc, Baris (998) considr a coninuous marials flow producion sysm wih mulipl machins in sris bu no inrmdia buffrs. Howvr, machins procssing im is drminisic [5]. Som rsarch focuss on ohr prformanc masurs, such as producion ra and du-im prformanc. Jacobs and Mrkov (995) prformd sysm horic analysis of du-im prformanc in producion sysms [6]. Tan and Yralan (997) proposs a dcomposiion modl for coninuous marials flow producion sysms o valua producion ra in sady sa [7]. Li and Mrkov (995) valuas hroughpu in srial producion lins wih non-xponnial machins [8]. Howvr, propris of sling im and producion loss rciv lss anion.
3 I is imporan o xnd h ransin analysis o srial producion lins wih ohr machin rliabiliy modls, i.. xponnial and no-xponnial (wibull, gamma, lognormal). Howvr, h machin rliabiliy modl is much mor difficul o idnify. Sricly spaking, i rquirs h idnificaions of h hisograms of up- and downim, which rquir a vry larg numbr of masurmns during a long priod of im. Th rsul is ha h machins ral rliabiliy modl on h facory floor ar, pracically, nvr known. Ralisically spaking, machins avrag up- and downim (T up and T down ) and h cofficin varianc of up- and downim (CV up and CV down ) may b h only characrisics of rliabiliy modls availabl from h facory floor. Thrfor, i is criical o invsiga h impacs of h machin paramrs, such as T up, T down, CV up, CV down, Efficincy (), and buffr siz (N) o h producion lins ransin prformancs, such as producion ra, sling im, oal producion, and producion loss. Prvious rsarch mosly focus on h wo or hr machin sysm o rduc h sysm complxiy. For insanc, ransin bhavior of wo-machin lins wih Gomric rliabiliy was sudid by Mrkov al. (2) [9]. Baris and Sanly (29) analyzs gnral Markovian coninuous marials flow producion sysms wih wo machins [2]. Bruno (2) considrs a fluid sysm wih wo machins whos sas ar Markovian and a fini buffr bwn hm [2]. im al. (22) providd an uppr bound for carrirs in a hr-machin srial producion lin [22]. Currnly hr ar no analyical ools o addrss h problms wih mulipl machins and buffrs during ransin priods, his dissraion uss simulaion wih C++ programming sudy h mulipl machins and buffrs (up o ) and illusras h ransin prformanc by cas sudis.
4.2 Oulin Th oulin of his dissraion is as follows: Chapr 2 inroducs machins rliabiliy modls and h sysm modls considrd in his hsis. Chapr 3 and Chapr 4 invsiga h ffcs of sysm paramrs (including machin fficincy, raio of N and T down (), Machins avrag downim T down, and cofficin of variaion CV) on diffrn prformanc masurs, including sling im of producion ra ( spr ), sling im of work-in-procss ( swip ), oal producion (TP), producion loss (PL). Chapr 3 invsigas h ransins of srial producion lin wih machins rliabiliy modl saisfying xponnial disribuion. Chapr 4 xplors h ransin prformancs for,, Log-Normal producion lins, rspcivly. Finally, h conclusions and opics for fuur rsarch ar providd in Chapr 5.
5 Chapr 2 Sysm Modls and Problms In ordr o formaliz h sysm modling and problms, his chapr dfins a s of sandard vocabulary usd hroughou his hsis [23]. 2. Trminology Srial producion lin: Srial producion lin a group of producing unis, arrangd in conscuiv ordr, and marial handling dvics ha ranspor pars (or jobs) from on producing uni o h nx. Figur shows h block diagram of a srial producion lin whr m i, i= M, rprsn producing unis and b i, i= M-, ar marial handling dvics. m b m 2 m M- b M- m M Figur Srial producion lin Cycl im (τ) : h im ncssary o procss a par by a machin. Th cycl im may b consan, variabl, or random. In larg volum producion sysms, is pracically always consan or clos o bing consan. This is h cas in mos producion sysms of h auomoiv, lcronics, applianc, and ohr indusris. Variabl or random cycl im aks plac in job-shop nvironmns whr ach par may hav diffrn procssing spcificaions. In his rsarch, w considr only machins wih a consan cycl im;
6 howvr, similar dvlopmns can b carrid ou for h cas of random (.g., xponnially disribud) procssing im. Machin capaciy c: h numbr of pars producd by a machin pr uni of im whn h machin is up. Clarly, in h cas of consan, Machins in a producion sysm may hav idnical or diffrn cycl ims. In h cas of idnical cycl im, h im axis may b considrd as slod or unslod. Slod im: h im axis is slod wih h slo duraion qual o h cycl im. In his cas, all ransiions - changs of machins saus (up or down) and changs of buffrs occupancy - ar considrd as aking plac only a h bginning or h nd of h im slos. Unslod im or coninuous im: changs of machins saus (up or down) and changs of buffrs occupancy may occur a any im momn. If h cycl ims of all machins ar idnical, such a sysm is rfrrd o as synchronous. If h cycl ims ar no idnical, h sysm is calld asynchronous. Producion sysms wih machins having diffrn cycl ims ar ypically considrd as opraing in unslod im. In h unslod cas, producion sysms can b concpualizd as ypically considrd as discr vn sysms or as flow sysms. Discr vn sysm: a job (i.. par) is ransfrrd from h producing machin o h subsqun buffr (if i is no full) only afr h procssing of h whol job is compl. In his cas, h buffr occupancy is a non-ngaiv ingr.
7 Flow sysm: infinisimal pars of h job ar (concpually) ransfrrd from h producing machin o h subsqun buffr if i is no full. Similarly, an infinisimal par of a job is akn by a downsram machin from h buffr, if h machin is no down and h buffr is no mpy. In his cas, hr is a coninuous flow of pars ino and from h buffrs. Clarly, h buffr occupancy in his siuaion is a non-ngaiv ral numbr. Flow sysms ar somims asir o analyz and ofn lad o rasonabl conclusions. Machin rliabiliy modl: h probabiliy mass funcions (pmf s) or h probabiliy dnsiy funcions (pdf s) of h up- and downim of h machin in h slod or unslod im, rspcivly. In addiion, h xpcd valu and cofficin of variaion of up- and downim ar dnod as T up, T down, CV up and CV down, rspcivly. 2.2 Machin Rliabiliy Modls In his dissraion, h following four machin rliabiliy modls ar usd: Exponnial,, and. In h coninuous im cas, ach machin is dnod as [ ( ) ( )], whr [ ( ) ( )], ar h pdf s of up- and downim, rspcivly. Th xpcd valu and cofficin of variaion of up- and downim, T up, T down, CV up and CV down, rspcivly ar shown in Tabl. Exponnial rliabiliy modl (xp): Considr a machin in Figur 2, which is a coninuous im analogu of h gomric machin. Namly, if i is up (rspcivly, down) a im, i gos down (rspcivly, up) during an infinisimal im wih probabiliy (rspcivly, ). Th paramrs and ar calld h brakdown and rpair ras, rspcivly.
8 o( ) o( ) o( ) o( ) Figur 2 Exponnial Rliabiliy Modl I can b shown ha h pdf's of h up- and downim of his machin, dnod as up and down, ar as follows: Clarly, up and down ar xponnial random variabls and w rfr o such a machin as an xponnial machin, i.., obying h xponnial rliabiliy modl. In addiion, i is asy o show ha for an xponnial machin rliabiliy modl (W): disribuion is widly usd in Rliabiliy Thory. For a machin obying rliabiliy modl, is up- and downim pdf s ar givn by ( ) ( ) ( ) ( ) whr and ar posiiv numbrs. I can b calculad ha for a machin
9 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) rliabiliy modl (ga): For a machin obying h gamma rliabiliy modl, is up- and downim pdf s ar givn by gamma disribuion, ( ) ( ) ( ) ( ) ( ) ( ) Whr, ( ) and and ar posiiv numbrs. In addiion, i can b calculad ha for a gamma machin rliabiliy modl (LN): For a machin obying h log-normal rliabiliy modl, is up- and downim pdf's ar givn by ( ) ( ) ( ) ( )
Whr, and ar posiiv numbrs. In addiion, i can b calculad ha for a lognormal machin Tabl Expcaion, varianc, and cofficins of variaion of coninuous random variabls Random variabl Expcaion Varianc CV Exponnial ( ) [ ( ) ( ) ] ( ) ( ) ( ) ( ) 2.3 Sysms Considrd 2.3. Coninuous Srial Producion Lins Coninuous srial producion lins ar illusrad in Figur 3 whr circls rprsn machins and rcangls rprsn buffrs. m b m 2 m M- b M- m M Figur 3 Srucural Modl
Convnions: (a) Blockd bfor srvic. (b) Th firs machin is nvr sarvd; h las machin is nvr blockd. (c) Flow modl, i.., infinisimal quaniy of pars, producd during δ, ar ransfrrd o and from h buffrs. (d) Th sa of ach machin (up or down) is drmind indpndnly from all ohr machins. () Tim-dpndn failurs. In coninuous im cas, srial producion lins shown by Figur 3 opra according o h following assumpions: a) Th sysm consiss of M machins m i,, and M- buffrs,. b) Each machin m i,, has wo sas: up and down. Whn up, h machin is capabl of producing wih ra (pars/uni of im); whn down, no producion aks plac. c) Th up- and downim of ach machin ar coninuous random variabls, and,, and ar drmind by is rliabiliy modl. I is assumd ha hs random variabls ar muually indpndn. d) Each in-procss buffr is characrizd by is capaciy, ) Machin m i,, is sarvd a im if i is up a im and buffr is mpy a im.
2 f) Machin m i,, is blockd a im if i is up a im, buffr is full a im and machin m i+ fails o ak any work from his buffr a im. 2.3.2 Sysms Considrd In his dissraion, coninuous srial producion lins ar oprad according o h following assumpions: a) Th sysm consiss of M idnical machins m i,, and M- idnical buffrs,. b) Each machin m i,, has wo sas: up and down. Whn up, h machin is capabl of producing wih ra (pars/uni of im); whn down, no producion aks plac. Machins ar down iniially. c) Machins m i,, opra indpndnly and oby coninuous rliabiliy modl. d) Each buffr has fini capaciy, and is mpy iniially. ) Machin m i, i =2,...,M, is sarvd if i is up and buffr b i- is mpy. I is assumd ha machin m is nvr sarvd; f) Machin m i, i =,...,M-, is blockd if i is up, buffr b i has N i pars and machin m i+ fails o ak a par. I is assumd ha m M is nvr blockd. No ha coninuous srial producion lins wih hr or n idnical machins, whos rliabiliy modl saisfis Exponnial,, and, rspcivly, ar rsarchd in his dissraion. Thrfor, flow modl is usd in h rsarch. As a rsul, h sa of h buffr is a ral numbr bwn and N. Sinc simulaion mhod is usd in his work, machins cycl ims ar s o minu. In ohr words, ach
3 machin s capaciy c is par/min. Tim axis is dividd ino svral im infinisimal slos δ. δ is s o.5 minu. 2.3.3 Sysm Paramrs, : Machins avrag up- and downim., : Cofficin of variaion of up- and downim. : machin fficincy, which is h xpcd valu of h numbr of pars producd during a cycl im. In his cas, is dmonsrad by quaion blow: N: buffr capaciy, h maximum numbr of pars ha h buffr can sor. I is assumd hroughou ha N <, implying ha buffrs ar fini. Th numbr of pars conaind in a buffr a a givn im is rfrrd o as is occupancy. Sinc in a producion sysm, h occupancy of a buffr a a givn im (slo or momn) dpnds on is occupancy a h prvious im (slo or momn), buffrs ar dynamical sysms wih h occupancy bing hir sas. If h machins ar modld as discr vn sysms, h sa of h buffr is an ingr bwn and N. In flow modls, sas ar ral numbrs bwn and N. : Raio of and. Th largr is, h mor procion o machins from sarvaion and blockag producd by buffrs.
4 2.4 Prformanc Masurs Th following prformanc masurs ar considrd: Producion ra (PR): avrag numbr of pars producd by h las machin of a producion sysm pr cycl im in h ransin sa of sysm opraion. Toal work-in-procss (WIP): avrag numbr of pars conaind in all h buffrs of a producion sysm in h ransin sa of is opraion. Toal producion (TP) : avrag oal numbr of pars producd by h las machin in h im duraion T. ( ) quanifis how much producion will b gaind in im T. Producion loss (PL): h prcnag of rducd producion from h bginning o im T compard wih oal producion in sady sa. [ ( )] Whr, is h producion ra in h sady sa of sysm opraion. PL quanifis h prcnag of producion loss du o ransin procss. Sling im of producion ra( ): h xpcd im ncssary for o rach and rmain wihin ±5% of. masurs how fas h sysm nrs sady sa rgarding producion ra. Sling im of work-in-procss( ): h xpcd im ncssary for o rach and rmain wihin ±5% of ; masurs how fas h sysm nrs sady sa rgarding work-in-procss.
5 In his work, w analyz oal producion and producion loss during a shif of duraion T cycls. W assum ha T = 5 minus, which is ypical for auomoiv assmbly plans whr h cycl im is around minu and h shif is 8 hours. T is s o 5 minus in all h sysms invsigad in his hsis o simula a ypical auomoiv assmbly plans. 2.5 Problm Samn Prvious rsarch of ransin prformanc ar limid o som non-exponnial producion lins, and h analysis ar mainly focusd on h ffcs of cofficin varianc (CV) o h producion ras (PR). I is shown ha h producion ra is monoonically dcrasing funcion of cofficin variaion [23]. I is imporan o xnd hs ransin analysis o ohr machin modls, and furhrmor o invsiga h gnral ffcs of sysm paramrs on ohr prformanc masurs.. Exnd hs ransin analysis o ohr machin modls. This dissraion is dvod o invsiga h ffcs of sysm paramrs on prformanc masurs for ransin srial producion lin wih ohr machin rliabiliy modls. Th rliabiliy modls invsigad in his dissraion includ xponnial and noxponnial (,, ). 2. Invsiga h gnral ffcs of sysm paramrs on ohr prformanc masurs. In a ral producion lin sysm, machin rliabiliy modls ar much mor difficul o idnify. Thrfor, i is of significan pracical imporanc o invsiga h gnral ffcs of sysm paramrs on prformanc masurs for ransin srial producion lin. This dissraion invsigas h ffcs of sysm paramrs (including,, T down and CV) on ohr prformanc masurs,
6 including sling im of producion ra ( ), sling im of work-in-procss ( ), oal producion (TP), producion loss (PL). Th rlaionship bwn h prformanc masurs and sysm paramrs rvals h fundamnal principls ha characriz h bhavior of such sysms and can b usd as a guidlin for produc lins managmn and improvmn.
7 Chapr 3 Exponnial Sysms This chapr invsigas ransins of xponnial srial lin. Firs, ransins of individual xponnial machin and buffrlss xponnial srial lin ar analyzd. Scond, producion ra of xponnial lin is approximad by using gomric lin. Third, h ffcs of sysm paramrs,, and T down, of xponnial srial producion lin on h ransin prformanc masur which ar sling im, oal producion and producion loss ar analyzd. Th xponnial srial producion lin which is oprad undr h assumpions (a-f) in scion 2.3.2. Th paramrs of machins rliabiliy modl ar drmind by h sysm paramrs abl in ach scion. 3. Transins of Exponnial Lins 3.. Transins of Individual Exponnial Machin L x i (), i { = down, = up} b h probabiliy ha h machin is in sa i a im. Thn, h voluion of x() = [x () x ()] T can b dscribd a Markov chain: x ( ) Ax( ), x() + x() =, A. Th ignvalus of marix A ar and (λ + μ) and h corrsponding ignvcors ar:, L
8. Q Thn,, Q h voluion of h sysm sa can b calculad as:. () () () () () () () () () () ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( x x x x x x x x x x Q Q x x Sinc λ and μ ar boh posiiv, ) ( nds o as approachs infiniy, and, hrfor,, ) ( () ) ( ) ( () ) ( ) ( ) ( ) ( ) ( x x x x x x x x whr. ) (, ) ( down up up down up down T T T x T T T x Clarly, h ransin of an individual xponnial machin is characrizd by ) Th disanc bwn h iniial condiion and h sady sa; and 2) Sysm mod ( ).
9 In addiion, if h machin is iniially in h sady sa, i.., x ), x (), ( hn i rmains in h sady sa for all : x ), x ( ). ( An illusraion is givn in Figur 4 for an xponnial machin wih T up = 8, T down = 2 (i.., λ =.25 and μ =.5). (a) Machin iniially down (b) Machin iniially up Figur 4 Transins of machin sa of an individual xponnial machin (T up = 8, T down = 2) 3..2 Transins of Buffrlss Exponnial Srial Lin Considr an xponnial srial lin having all buffrs wih zro capaciy. Clarly, h producion ra of his lin a im is givn by: PR() = Pr[all machins ar up a im ]. L x i, j (), i { = down, = up}, j {, 2,, M} b h probabiliy ha h machin m j is in sa i a im. Thn,
2 M M ( i i ) PR( ) x, i ( ) i ( x, i () i ). i Th sady sa producion ra is i M i i PR( ) and h ransin of PR() conains a numbr of mods dfind by all possibl combinaions of h machins in h sysm. An illusraion is givn in Figur 5 for a buffrlss fiv-machin xponnial lin wih idnical machins (T up = 8, T down = 2). Th machins ar assumd o b down iniially. In Figur 5 (a), w plo h ransins of h sysm hroughpu ra, PR() (which is qual o CR() for buffrlss lins). To compar h ransins of sysm prformanc wih individual machin, w plo h ransins of h probabiliy ha a machin is up in Figur 5(b) along wih PR(). In addiion, w normaliz boh rms by hir corrsponding sady sa valus. As on can s from h figur, du o h inracion of h machins in h sysm, h ransins of h sysm prformanc is slowr han hos of individual machins. Similar obsrvaion can b mad whn h iniial condiion is changd (s Figur 6). (a) PR() and CR() (b) Normalizd prformanc Figur 5 Transins of buffrlss fiv-machin xponnial lin wih idnical machins (T up = 8, T down = 2, machins iniially down)
2 I should b nod ha, whn h machins ar iniially up, h sysm hroughpu, in fac, bnfis from h ransins as i nvr nrs blow is sady sa lvl. For gnral iniial condiions, h sysm hroughpu should b slowr han h machin wih h slows ransins. (a) PR() and CR() (b) Normalizd prformanc Figur 6 Transins of buffrlss fiv-machin xponnial lin wih idnical machins (T up = 8, T down = 2, machins iniially up) 3..3 Exponnial vs. Gomric I is wll known ha gomric disribuion is h discr counrpar o h xponnial disribuion. Th gomric rliabiliy modl is dfind as follows: L s(n) { = down, = up} dno h sa of a machin during cycl im n. Thn, h ransiion probabiliis ar givn b: Pr[ s( n ) s( n) ] P,Pr[ s( n ) s( n) ] P, Pr[ s( n ) s( n) ] R,Pr[ s( n ) s( n) ] R, whr P and R ar rfrrd o as h brakdown and rpair probabiliis, rspcivly. Clarly, h up- and downim of a machin wih h rliabiliy modl abov ar gomric random variabls wih man T up and T down givn by:
22 T up, T P down. R Th main diffrnc bwn h gomric rliabiliy modl and h xponnial modl is ha, a gomric machin opras undr a slod im axis (i.., in discr im) wih h slo duraion qual o is cycl im and all vns (machin brakdown/rpair, ransporaion of pars, c.) ak plac ihr a h bginning or a h nd of a im slo, whil an xponnial machin opras in coninuous im and an vn can ak plac a any im insan. In addiion, h flow modl considrd for h xponnial cas allows infinisimal pars o ravl wihin h sysm, whil h gomric cas only movs whol pars around. Dspi hs diffrncs, h wo modls ar vry similar. I can b shown ha producion lins wih gomric machins ar characrizd by discr-im discr-spac Markov chains. Analyical sudis hav bn carrid ou o invsiga h ransin bhavior of such sysms (s []) and an analyical procdur basd on rcursiv aggrgaion has bn dvlopd o approxima h ransin prformanc of a gomric lin wih high accuracy. On h ohr hand, producion lins wih xponnial machins ar characrizd by coninuous-im mixd-spac Markov procss, which is much mor difficul o sudy analyically. Sinc boh sysms shar a numbr of similariis, i bcoms inrsing o s if i is possibl o sudy h ransins of srial lins wih xponnial machins by ransforming h sysm ino on wih gomric machins. This ransformaion is, indd, vry sraighforward: Considr a srial lin wih xponnial machins dfind by scion 2.3.2, hn is corrsponding srial lin wih gomric machins ar givn by: go Pi i, Ri i, Ni Ni,
23 whr P i and R i ar h brakdown and rpair ras of machin mi in h gomric lin and N go i is h capaciy of buffr bi in h gomric lin. L PR go (n) dno h producion ra of h gomric lin during im slo n. Thn, w may approxima h producion ra of h original xponnial lin using: PR app go go PR PR go ( ) PR. In ohr words, a ingr im insans (i.., =, 2, 3 ), h producion ra of h xponnial lin is approximad using h producion ra of h gomric lin during h sam im slo. For non-ingr im insans, h producion ra of h xponnial lin is approximad using linar inrpolaion of h producion ras of h gomric lin during h nars wo im slos. An illusraion of his approximaion is providd in Figur 7. As on can s, h gomric lin-basd formula has vry good accuracy in approximaing h ransin producion ra of an xponnial lin. (a) PR() and PR app () (b) Approximaion rror Figur 7. Approximaion of PR() of fiv-machin xponnial lin wih idnical machins and idnical buffrs (T up = 4, T down =, N =, all machins iniially down, all buffrs iniially mpy)
24 Similarly, approximaion formulas for ohr ransin prformanc masurs ar proposd as follows: WIP ST CR app app i BL app app app i go i ( ) CR ( ) WIP ( ) ST ( ) BL go go i go i go go CR CR, go go CR CR ( ), i,..., go go STi STi ( ), i 2,..., M, go go BLi BLi ( ), i,..., M, app go i M, As an illusraion, w sudy h approximaion of hs ransin prformanc masurs for h sam fiv-machin lin considrd abov. Th rsuls ar summarizd in Figur 8- Figur. As on can s h accuracy of consumpion ra approximaion is similar o ha of h producion ra simaion. Th accuracy of work-in-procss approximaion is lowr bu sill wihin 5% of h buffr capaciy in mos cass. Th accuracy of sarvaion and blockag approximaion is similar, ypically wihin ±.2. (a) CR() and CR app () (b) Approximaion rror Figur 8 Approximaion of CR() of fiv-machin xponnial lin wih idnical machins and idnical buffrs (T up = 4, T down =, N =, all machins iniially down, all buffrs iniially mpy)
Figur 9 Approximaion of WIP i () of fiv-machin xponnial lin wih idnical machins and idnical buffrs (T up = 4, T down =, N =, all machins iniially down, all buffrs iniially mpy) 25
Figur Approximaion of ST i () of fiv-machin xponnial lin wih idnical machins and idnical buffrs (T up = 4, T down =, N =, all machins iniially down, all buffrs iniially mpy) 26
Figur Approximaion of BL i () of fiv-machin xponnial lin wih idnical machins and idnical buffrs (T up = 4, T down =, N =, all machins iniially down, all buffrs iniially mpy) 27
28 3.2 Transin Prformanc Analysis To analyz h sysm s ransin prformanc, sling ims, spr and swip, on has o know h bhavior of PR and WIP as a funcion of. Thrfor, in his scion, w firs analyz h rajcoris of PR() and WIP() and hn uiliz hm o valua h sling im. Th sysm paramrs in h simulaion ar shown in Tabl 2. Tabl 2 Exponnial Sysm Transin Prformanc Analysis Valus Paramrs M T down Rang [.7,.9] [3,] [,5] [3,9] Dfaul Valu.9 [3,] 3 5 Basd on sysm characrisics and simulaion rsuls in h following subscions, w hav h following conjcurs: ) PR approachs sady sa valu fasr han WIP in h sam sysm; 2) PR approachs sady sa valu fasr whn incrass. In conras, WIP nds longr im o rach h sady sa as incrass. 3) PR and WIP approachs sady sa slowr whn hr ar mor machins in h sysm; 4) Th diffrnc of h ransin ims bwn PR and WIP also bcoms mor significan as h machin numbr incrass. 5) PR and WIP approachs sady sa valu slowr whn incrass; 6) PR and WIP approachs sady sa valu slowr whn T down incrass; 3.2. Effcs of In ordr o analyz h ffcs of fficincy () on h ransin prformanc. PR and WIP ar simulad wih h following sysm paramrs:
Normalizd Oupu Normalizd Oupu Normalizd Oupu Normalizd Oupu 29 Tabl 3 Exponnial Sysm Paramrs (Effcs of on ransin prformanc) Paramrs M T down Valu [.7,.9] [3, ] 3 5 Analysis of PR and WIP: To compar h ransins of PR and WIP, Figur 2 shows h graphs of PR/PR ss and WIP/WIP ss for various and M and h following conjcurs ar obsrvd: ) Th producion ra (PR) approachs sady sa valu fasr han h work-inprocss (WIP) in h sam sysm. As bcoms largr, h diffrnc bcoms mor pronouncd. For xampls, a M=3, =.7, h sing im of PR and WIP ar around 8 and 5, rspcivly. A M=3, =.9, h sing im of PR and WIP ar around 5 and 25, rspcivly. M =.7 =.9 PR/PRss WIP/WIPss PR/PRss WIP/WIPss.8.8 3.6.6.4.4.2.2 5 5 2 25 3 35 4 45 5 5 5 2 25 3 35 4 45 5 PR/PRss WIP/WIPss PR/PRss WIP/WIPss.8.8.6.4.6.4.2.2 5 5 2 25 3 35 4 45 5 5 5 2 25 3 35 4 45 5 Figur 2 Effcs of on Transin Prformanc of Exponnial Lins
Normalizd Oupu Normalizd Oupu 3 2) PR approachs sady sa valu fasr whn incrass. In conras, WIP nds longr im o rach h sady sa as incrass. 3) PR and WIP boh approach sady sa slowr whn hr ar mor machins in h sysm. For insanc, whn hr ar hr machins in h sysms, machins fficincy is.7, h sling im of PR is clos o 2. In conras, h sling im of PR is mor han 5 whn hr ar n machins in h sysm. 4) Th diffrnc of h ransin ims bwn PR and WIP also bcoms mor significan as h machin numbr incrass. 3.2.2 Effcs of In ordr o analyz h ffcs of on h ransin prformanc. PR and WIP ar simulad wih h following sysm paramrs: Tabl 4 Exponnial Sysm Paramrs (Effcs of on ransin prformanc) Paramrs M T down Valu.9 [3, ] [,5] 5 Analysis of PR and WIP: To compar h ransins of PR and WIP, Figur 3 shows h M = =5 PR/PRss WIP/WIPss PR/PRss WIP/WIPss.8.8 3.6.4.6.4.2.2 5 5 2 25 3 35 4 45 5 5 5 2 25 3 35 4 45 5
Normalizd Oupu Normalizd Oupu Normalizd Oupu Normalizd Oupu 3 PR/PRss WIP/WIPss PR/PRss WIP/WIPss.8.8.6.4.6.4.2.2 5 5 2 25 3 35 4 45 5 5 5 2 25 3 35 4 45 5 Figur 3 Effcs of on Transin Prformanc of Exponnial Lins graphs of PR/PR ss and WIP/WIP ss for various and M, and h following conjcurs is obsrvd: ) PR and WIP boh approach sady sa valu slowr whn incrass. 3.2.3 Effcs of T down In ordr o analyz h ffcs of T down on h ransin prformanc. PR and WIP ar simulad wih h following sysm paramrs: Tabl 5 Exponnial Sysm Paramrs (Effcs of T down on ransin prformanc) Paramrs M T down Valu.9 [3, ] 3 [3, 9] Analysis of PR and WIP: To compar h ransins of PR and WIP, Figur 4 shows h M T down =3 T down =9 PR/PRss WIP/WIPss PR/PRss WIP/WIPss.8.8.6.6 3.4.4.2.2 5 5 2 25 3 35 4 45 5 5 5 2 25 3 35 4 45 5
Normalizd Oupu Normalizd Oupu 32 PR/PRss WIP/WIPss PR/PRss WIP/WIPss.8.8.6.4.6.4.2.2 Figur 4 Effcs of T down on Transin Prformanc of Exponnial Lins graphs of PR/PR ss and WIP/WIP ss for various and M, h following conjcur is obsrvd: 5 5 2 25 3 35 4 45 5 ) PR and WIP boh approach sady sa valu slowr whn T down incrass. 3.3 Sling Tim 5 5 2 25 3 35 4 45 5 Sling im of PR or WIP masurs how fas h sysm nrs sady sa in rms of PR or WIP. Th shorr sling im, h fasr h sysm approachs sady sa. To analyz h ffcs of,, and T down on h sling im of PR ( spr ) and WIP ( swip ), simulaions ar implmnd and h sysm paramrs ar shown in ach subscion. 3.3. Effcs of In ordr o analyz h ffcs of fficincy () on h sling im spr and swip, simulaions ar implmnd wih h following sysm paramrs: Tabl 6 Exponnial Sysm Paramrs (Effcs of on spr and swip ) Paramrs M N Valu [.,.2,.3,.4,.5,.6,.7,.8,.9,.95] [3, ] 3 [5,, 5] Analysis of spr and swip : Figur 5 shows h graphs of spr and swip vs. for various N and M.
spr swip spr swip spr swip 33 ) PR has shorr sling im han WIP in h sam sysm ( spr< swip ). 2) spr bcoms shorr as incrass and h slop bcoms largr as N incrass. In conras, swip is a convx funcion of. 3) spr and swip boh incras if hr ar mor machins in h sysm. spr swip M=3, N=5 2 8 6 4 2 = =2 =3 =4 =5 6 5 4 3 = =2 =3 =4 =5 8 6 2 4 2..2.3.4.5.6.7.8.9..2.3.4.5.6.7.8.9 9 8 7 = =2 =3 =4 =5 9 8 7 = =2 =3 =4 =5 M=3, N= 6 5 4 6 5 4 3 3 2 2..2.3.4.5.6.7.8.9..2.3.4.5.6.7.8.9 M=3, N=5 9 8 7 6 5 4 = =2 =3 =4 =5 2 8 6 4 2 8 = =2 =3 =4 =5 3 6 2 4 2..2.3.4.5.6.7.8.9..2.3.4.5.6.7.8.9
spr swip spr swip spr swip 34 M=, N=5 9 8 7 6 5 4 = =2 =3 =4 =5 3 25 2 5 = =2 =3 =4 =5 3 2 5..2.3.4.5.6.7.8.9..2.3.4.5.6.7.8.9 M=, N= 3 25 2 5 = =2 =3 =4 =5 5 45 4 35 3 25 2 5 = =2 =3 =4 =5 5 5..2.3.4.5.6.7.8.9..2.3.4.5.6.7.8.9 5 45 4 35 = =2 =3 =4 =5 5 45 4 35 = =2 =3 =4 =5 M=, N=5 3 25 2 5 3 25 2 5 5 5..2.3.4.5.6.7.8.9..2.3.4.5.6.7.8.9 Figur 5 Effcs of on spr and swip of Exponnial Lins 3.3.2 Effcs of In ordr o analyz h ffcs of on h sling im spr and swip, simulaions ar implmnd wih h following sysm paramrs: Tabl 7 Exponnial Sysm Paramrs (Effcs of on spr and swip ) Paramrs M N Valu [.6,.7,.8,.9,.95] [3, ] [, 2, 3, 4, 5] [5,, 5]
spr swip spr swip spr swip 35 Analysis of spr and swip : Figur shows h graphs of spr and swip vs. for various N and M. ) dos no hav significan impac on spr.. 2) has posiiv impac on swip, largr lads o a longr swip. For largr M, has mor significan linar impac on swip. spr swip M=3 N=5 35 3 25 2 =.6 =.7 =.8 =.9 =.95 2 8 6 4 2 =.6 =.7 =.8 =.9 =.95 5 8 6 4 5.5 2 2.5 3 3.5 4 4.5 5 2.5 2 2.5 3 3.5 4 4.5 5 7 =.6 4 =.6 6 =.7 =.8 =.9 35 =.7 =.8 =.9 M=3 N= 5 4 =.95 3 25 2 =.95 3 5 2.5 2 2.5 3 3.5 4 4.5 5 5.5 2 2.5 3 3.5 4 4.5 5 9 =.6 7 =.6 8 7 =.7 =.8 =.9 =.95 6 =.7 =.8 =.9 =.95 M=3 N=5 6 5 4 3 2 5 4 3 2.5 2 2.5 3 3.5 4 4.5 5.5 2 2.5 3 3.5 4 4.5 5
spr swip spr swip spr swip 36 M= N=5 4 3 2 =.6 =.7 =.8 =.9 =.95 5 45 4 35 =.6 =.7 =.8 =.9 =.95 3 25 9 2 8 5 7.5 2 2.5 3 3.5 4 4.5 5.5 2 2.5 3 3.5 4 4.5 5 M= N= 28 26 24 22 =.6 =.7 =.8 =.9 =.95 9 8 7 6 =.6 =.7 =.8 =.9 =.95 2 5 8 4 3 6.5 2 2.5 3 3.5 4 4.5 5 2.5 2 2.5 3 3.5 4 4.5 5 M= N=5 44 42 4 38 36 34 =.6 =.7 =.8 =.9 =.95 6 4 2 =.6 =.7 =.8 =.9 =.95 32 8 3 6 28 26 4 24.5 2 2.5 3 3.5 4 4.5 5 2.5 2 2.5 3 3.5 4 4.5 5 Figur 6 Effcs of on spr and swip of Exponnial Lins 3.3.3 Effcs of T down In ordr o analyz h ffcs of T down on h sling im spr and swip, simulaions ar implmnd wih h following sysm paramrs:
spr swip spr swip spr swip 37 Tabl 8 Exponnial Sysm Paramrs (Effcs of T down on spr and swip ) Paramrs M T down Valu [.6,.7,.8,.9,.95] 3 [, 3, 5] [5, 7, 9,, 3 5] Analysis of spr and swip : Figur 7 shows h graphs of spr and swip vs. T down for various and M. ) T down has posiiv impac on boh spr and swip, largr lads o a longr spr and swip. Th rlaionship is clos o linar. spr swip =.6 45 =.6 9 =.7 =.8 4 =.7 =.8 M=3 = 8 7 6 =.9 =.95 35 3 25 2 =.9 =.95 5 5 4 3 5 2 5 6 7 8 9 2 3 4 5 Tdown 5 6 7 8 9 2 3 4 5 Tdown M=3 =3 3 25 2 5 =.6 =.7 =.8 =.9 =.95 5 =.6 =.7 =.8 =.9 =.95 5 5 M=3 =5 5 6 7 8 9 2 3 4 5 Tdown 5 45 4 35 3 25 2 5 5 =5 =.6 =.7 =.8 =.9 =.95 5 6 7 8 9 2 3 4 5 Tdown 35 3 25 2 5 5 =.6 =.7 =.8 =.9 =.95 5 6 7 8 9 2 3 4 5 Tdown 5 6 7 8 9 2 3 4 5 Tdown Figur 7 Effcs of T down on spr and swip of Exponnial Lins
38 3.4 Toal Producion Toal producion (TP) dscribs how many producs can b producd in h producion duraion. TP is on of h mos imporan indics in sysm prformanc valuaion. Alhough PR in ransin priod can b highr han PR SS occasionally, TP during ransin will b smallr han oal producion undr h sam sysm in sady sas. Th analysis in his scion will invsiga h impac on TP du o sysm paramrs. According o scion 4.2., mos rsarch sysms rach sady sa during im slo T which is qual o 5 if is largr han.5, hrfor, w analyz h sysms wih machin fficincy largr han.5. 3.4. Effcs of In ordr o analyz h ffcs of fficincy () on TP, simulaions ar implmnd wih h following sysm paramrs: Tabl 9 Exponnial Sysm Paramrs (Effcs of on TP) Paramrs M T down Valu [.5,.55,.6,.65,.7,.75,.8,.85,.9,.95] [3, ] 3 [5,, 5] Analysis of TP: Figur 8 shows h graphs of TP vs. for various T down and M. ) has posiiv impac on TP, largr lads o a largr TP, h rlaionship is clos o linar. 2) Th largr, h lss impac of on TP. If machins fficincy is.95, for insanc, hr is no significan diffrnc whn incrass from o 5.
TP TP TP TP TP TP 39 T down 5 5 M 3 45 45 4 4 35 35 3 3 25 25 2 =5 5.5.55.6.65.7.75.8.85.9.95 = =2 =3 =4 2 5 =5.5.55.6.65.7.75.8.85.9.95 = =2 =3 =4 5 45 45 4 4 35 35 3 3 25 25 2 =5 5.5.55.6.65.7.75.8.85.9.95 = =2 =3 =4 2 5 =5.5.55.6.65.7.75.8.85.9.95 = =2 =3 =4 5 45 4 4 35 35 3 25 3 2 25 2 =5 5.5.55.6.65.7.75.8.85.9.95 = =2 =3 =4 5 =5 5.5.55.6.65.7.75.8.85.9.95 = =2 =3 =4 Figur 8 Effcs of on Toal Producion of Exponnial Lins
TP TP TP TP 4 3) TP dcrass if hr ar mor machins in h sysm. 3.4.2 Effcs of In ordr o analyz h ffcs of on TP, simulaions ar implmnd wih h following sysm paramrs: Tabl Exponnial Sysm Paramrs (Effcs of on TP) Paramrs M T down Valu [.6,.7,.8,.9,.95] [3, ] [, 2, 3, 4, 5] [5,, 5] Analysis of TP: Figur 9 shows h graphs of TP vs. for various T down and M. ) has posiiv impac on TP, largr lads o a largr TP. Howvr, TP sauras around =2. T down 5 5 45 4 M 3 =.6 =.7 =.8 =.9 =.95 45 4 35 =.6 =.7 =.8 =.9 =.95 35 3 3 25 25 2 2 5 5.5 2 2.5 3 3.5 4 4.5 5.5 2 2.5 3 3.5 4 4.5 5 5 45 4 =.6 =.7 =.8 =.9 =.95 45 4 35 =.6 =.7 =.8 =.9 =.95 35 3 3 25 25 2 2 5 5.5 2 2.5 3 3.5 4 4.5 5.5 2 2.5 3 3.5 4 4.5 5
TP TP TP TP 4 5 45 =.6 4 =.6 =.7 =.7 4 =.8 =.9 35 =.8 =.9 =.95 =.95 35 3 3 25 25 2 2 5 5.5 2 2.5 3 3.5 4 4.5 5.5 2 2.5 3 3.5 4 4.5 5 Figur 9 Effcs of on Toal Producion of Exponnial Lins In ohr words, h incras of TP from = o = 2 is significan; h impac on TP for >2 is ngligibl. 3.4.3 Effcs of Tdown In ordr o analyz h ffcs of T down on h TP, simulaions ar implmnd wih h following sysm paramrs: Tabl Exponnial Sysm Paramrs (Effcs of T down on TP) Paramrs M T down Valu [.6,.7,.8,.9,.95] [3, ] [, 3, 5] [5, 7, 9,, 3 5] 45 4 M 3 =.6 =.7 =.8 =.9 =.95 4 35 =.6 =.7 =.8 =.9 =.95 35 3 3 25 25 2 2 5 5 5 6 7 8 9 2 3 4 5 Tdown 5 6 7 8 9 2 3 4 5 Tdown
TP TP TP TP 42 3 45 4 =.6 =.7 =.8 =.9 =.95 45 4 35 =.6 =.7 =.8 =.9 =.95 35 3 3 25 2 25 5 2 5 6 7 8 9 2 3 4 5 Tdown 5 6 7 8 9 2 3 4 5 Tdown 5 45 4 =.6 =.7 =.8 =.9 =.95 45 4 35 =.6 =.7 =.8 =.9 =.95 35 3 3 25 2 25 5 2 5 6 7 8 9 2 3 4 5 Tdown 5 6 7 8 9 2 3 4 5 Tdown Figur 2 Effcs of T down on Toal Producion of Exponnial Lins Analysis of TP: Figur 2 shows h graphs of TP vs. T down for various and M. ) T down has ngaiv impac on TP, largr T down lads o smallr TP. Th rlaionship is linar. 2) Th maximum variaion of TP du o T down changs is lss han % a a fixd. 3.5 Producion Loss Producion loss (PL) is a masur of chang in oal producion comparing wih ha in sady sa. This raio conains informaion of rlaiv producion loss. Howvr, i dos
PL PL 43 no ransla dircly ino h valu of oal producion. Thrfor, low PL dos no ncssarily imply high TP. 3.5. Effcs of In ordr o analyz h ffcs of fficincy () on PL, simulaions ar implmnd wih h following sysm paramrs: Tabl 2 Exponnial Sysm Paramrs (Effcs of on PL) Paramrs M T down Valu [.5,.55,.6,.65,.7, [3, ] 3 [5,, 5].75,.8,.85,.9,.95] Analysis of PL: Figur 2 shows h graphs of PL vs. for various T down and M. ) has ngaiv impac on PL, largr lads o a smallr PL, h rlaionship is clos o linar. 2) Whn is qual, h mor machin, h largr producion loss. Machin fficincy is.95 whn hr ar machins in h sysm, for insanc, h producion loss could b as much as 4% if T down is 5. 3) PL incrass if hr ar mor machins in h sysm. T down 5..9.8.7 M 3 = =2 =3 =4 =5.25.2 = =2 =3 =4 =5.6.5.5.4.3..2..5.5.55.6.65.7.75.8.85.9.95.5.55.6.65.7.75.8.85.9.95
PL PL PL PL 44.2.8.6.4 = =2 =3 =4 =5.5.4 = =2 =3 =4 =5.2..3.8.6.2.4..2.5.55.6.65.7.75.8.85.9.95.5.55.6.65.7.75.8.85.9.95 5.2.8.6.4 = =2 =3 =4 =5.5.4 = =2 =3 =4 =5.2..3.8.6.2.4..2.5.55.6.65.7.75.8.85.9.95.5.55.6.65.7.75.8.85.9.95 Figur 2 Effcs of on Producion Loss of Exponnial Lins 3.5.2 Effcs of In ordr o analyz h ffcs of on PL, simulaions ar implmnd wih h following sysm paramrs: Tabl 3 Exponnial Sysm Paramrs (Effcs of on PL) Paramrs M T down Valu [.6,.7,.8,.9,.95] [3, ] [, 2, 3, 4, 5] [5,, 5] Analysis of PL: Figur 22 shows h graphs of PL vs. for various T down and M.
PL PL PL PL PL PL 45 ) has posiiv impac on PL, largr lads o a largr PL. Howvr, h slop dcrass as incrass, PL sauras a a crain poin whn incrass (Th largr, h smallr ). T down 5.65.6.55.5 M 3 =.6 =.7 =.8 =.9 =.95.35.3.25 =.6 =.7 =.8 =.9 =.95.45.2.4.35.5.3..25.2.5.5.5 2 2.5 3 3.5 4 4.5 5.5 2 2.5 3 3.5 4 4.5 5.3.2.. =.6 =.7 =.8 =.9 =.95.45.4.35 =.6 =.7 =.8 =.9 =.95.9.3.8.25.7.2.6.5.5.4..3.5 2 2.5 3 3.5 4 4.5 5.5.5 2 2.5 3 3.5 4 4.5 5 5.8.6.4 =.6 =.7 =.8 =.9 =.95.6.55.5.45 =.6 =.7 =.8 =.9 =.95.2.4..8.6.35.3.25.2.5.4.5 2 2.5 3 3.5 4 4.5 5..5 2 2.5 3 3.5 4 4.5 5 Figur 22 Effcs of on Producion Loss of Exponnial Lins
ProducionLoss ProducionLoss ProducionLoss ProducionLoss 46 3.5.3 Effcs of T down In ordr o analyz h ffcs of T down on h PL, simulaions ar implmnd wih h following sysm paramrs: Tabl 4 Exponnial Sysm Paramrs (Effcs of T down on PL) Paramrs M T down Valu [.6,.7,.8,.9,.95] [3, ] [, 3, 5] [5,6,7,8,9,,,2,3, 4,5,6,7,8, 9,2] Analysis of PL: Figur 23 shows h graphs of PL vs. T down for various and M. ) T down has posiiv impac on PL, largr T down lads o largr PL. I is clos o linar. Whn and M ar largr, h slop dcrass as T down incrass..8 M 3 =.6.4 =.6.7.6 =.7 =.8 =.9 =.95.35.3 =.7 =.8 =.9 =.95.5.4.25.2.5.3..2.5. 5 5 2 Tdown 5 5 2 Tdown 3.6.4.2 =.6 =.7 =.8 =.9 =.95.55.5.45.4 =.6 =.7 =.8 =.9 =.95..8.6.4.2 5 5 2 Tdown.35.3.25.2.5..5 5 5 2 Tdown
ProducionLoss ProducionLoss 47 5.22.2.8.6 =.6 =.7 =.8 =.9 =.95.6.55.5.45 =.6 =.7 =.8 =.9 =.95.4.2..8.6.4.2 5 5 2 Tdown.4.35.3.25.2.5. 5 5 2 Tdown Figur 23 Effcs of T down on Producion Loss of Exponnial Lins 3.6 Summary This chapr invsigas h ransin of an individual xponnial machin and buffrlss xponnial srial lin. Rsuls show ha h ransin of an individual xponnial machin is characrizd by wo facors. On is h disanc bwn h iniial condiion and h sady sa, h ohr is h sysm mod -(l+m). For buffrlss xponnial srial lin, in gnral condiion, h sysm hroughpu should b slowr han h machin wih h slows ransins. W invsiga if h ransins of srial lins wih xponnial machins could b ransformd ino h sysm wih gomric machins. Rsuls show ha h gomric lin-basd formula has vry good accuracy in approximaing h ransin producion ra of an xponnial lin. Th ransin srial producion lin wih machin rliabiliy modl saisfying xponnial disribuion ar also analyzd. Simulaions ar implmnd o analyz h ffcs of sysm paramrs, including,, and T down on h ransin prformanc sling im